Much of the modern art of investing relates to the evaluation of the appropriate trade-off of risk against return in the constitution of portfolios for investors. The problem is not simple. The investor provides unique combinations of investment goals and objectives, as well as risk perceptions that all naturally fluctuate over time based on past experiences and future expectations. Portfolios are typically baskets of instruments chosen from a universally diversified supermarket of investments. And, whereas return is easily quantified, much of the number crunching of modern portfolio theory now deals with the quantification of risk.
Risk refers to the uncertainty of the financial outcome of an investment portfolio following a given investment period. The value of the portfolio will fluctuate against a headwind of events that can never be foreknown, justifying modeling it as a random variable in a stochastic process. The uncertainty is practically nil, however, for certain offerings of assets such as short-term (30 days) treasury bills, yields of which are guaranteed by the taxation powers of government and their ability to expand the monetary float. Risk is thus better viewed in the context of a riskless benchmark: investors choose to invest in risky assets, and be subjected to their random fluctuations, on the condition they yield a risk premium, i.e. an incremental gain relative to the benchmark. Performance of the risky asset itself can also be benchmarked to its fluctuations: the greater the fluctuations, the greater the expected risk premium for a risk-adverse investor.
Early on, in formulating the expected returns-variance of returns rule leading to the constitution of efficient portfolios along an efficient frontier, H. M. Markowitz (see “Portfolio Selection”, Journal of Finance, March 1952, pp. 77-91) dealt with this uncertainty by suggesting that “if the term risk was replaced by variance of return (or standard deviation of return), little change in apparent meaning would result.” Unfortunately, some 50 years later, this suggestion has been popularized to the extent that risk has mostly become synonymous with variance or standard deviation, or volatility of return as measured by either, usually without the slightest reference to the original context of optimal efficient frontier. The work of H. M. Markowitz (see “Portfolio Selection”, Journal of Finance, March 1952, pp. 77-91) is classified as a linear/quadratic as well as an expectation/dispersion risk measure by G. C. Pflug (see “How to measure risk ?”, Modeling and Decisions in Economics: Essays in Honor of Franz Ferschl, Physica-Verlag, 1999). U.S. Pat. Nos. 6,003,018, 6,275,814 and 6,282,520 are centered on the concept of efficient frontier.
The paradox that volatility can also induce safe beneficial gains, not only risky detrimental losses, has led others to consider as more appropriate a class restricted to downside risk measures (see “A Brief History of Downside Risk Measures”, by David Nawrocki, Journal of Investing, Vol. 8, No. 3, Fall 1999, pp. 9-25), most notably the below-mean semi-variance, the below-target semi-variance (SVt), and the Lower Partial Moment (LPM), the latter due to Vijay S. Bawa (see “Optimal Rules for Ordering Uncertain Prospects”, Journal of Financial Economics, Vol. 2, No. 1, 1975, pp. 95-121) and Peter C. Fishburn (see “Mean-Risk Analysis with Risk Associated with Below Target Returns”, American Economic Review, Vol. 67, No. 2, 1977, pp. 116-126). Therein, the SVt, an expectation/dispersion risk measure (see “How to measure risk?”, G. C. Pflug, Modeling and Decisions in Economics: Essays in Honor of Franz Ferschl, Physica-Verlag, 1999), is shown to be a subclass of LPM. The LPM is related to the notion of moments of a probability density function in general. For a random variable x occurring with a probability density P(x), the nth moment about a point or target t over the full range of x is the weighted sum or integral, for discrete or continuous functions P(x), respectively, of the difference (x−t)n, the weights corresponding to the probability P(x). The LPM notion restricts the range of x to values below the target t while expanding the range of the exponent n from integer to real values a. The LPM then qualifies below-target risk in terms of the so-called risk tolerance parameter a. The LPM for (a=0) corresponds to the below-target probability, or the probability of loss given the target establishes the threshold in the profit and loss probability density and cumulative distribution functions. The LPM for (a=1) is the unconditional expected loss, whereas the LPM for (a=2) is the SVt. Basing risk tolerance on the lower partial moments of a stochastic distribution then bridges over to the field of ranking portfolios based on stochastic dominance and the values of their nth moment over a range of x (see “Stochastic Dominance: Investment Decision Making under Uncertainty”, H. Levy, Kluwer Academic Publishers, Boston, Mass., 392 pp. 1998).
A widespread downside risk measure is the value-at-risk measure V@Rα. V@Rα has gained regulatory status in defining minimum capital reserves or standards in relation to banks' market risk exposure, or risk of loss from adverse movements in the market values of assets, liabilities or off-balance-sheet positions (see “Basle Committee on Banking Supervision”, Basle, Amendment to the Capital Accord to Incorporate Market Risks, Federal Reserve System, 1996; “Conservatism, Accuracy and Efficiency: Comparing Value-at-Risk Models”, J. Engel and M. Gizycki, Working paper 2, Policy Development and Research, Australian Prudential Regulation Authority, Reserve Bank of Australia, March 1999; “Evaluation of Value-at-Risk Models Using Historical Data”, D. Hendricks, Federal Reserve Bank of New York Economic Policy Review, April 1996; “Bank Capital Requirements for Market Risk: the Internal Models Approach”, D. Hendricks and B. Hirtle, Federal Reserve Bank of New York Economic Policy Review, December 1997; and “Value-at-Risk: Recent Advances”, I. N. Khindanova and S. T. Rachev, Handbook on Analytic-Computational Methods in Applied Mathematics, CRC Press, 2000). V@Rα is also generously called upon in evaluating the performance of corporate pension plans, and has been built into the framework of many commercial software risk packages. V@Rα corresponds to the measure on the downside of the applied probability density function that is exceeded further on the downside only by a given very small probability of occurrences a, typically 1% or 5%. V@Rα is classified as an inverse-linear risk measure as it is linear in the inverse distribution function or quintile function α (see “How to measure risk?”, G. C. Pflug, Modeling and Decisions in Economics: Essays in Honor of Franz Ferschl, Physica-Verlag, 1999). Application of V@Rα to typical portfolios generally results in the highlighting of exceptionally high negative returns (losses). Surprisingly, no benchmark is set explicitly in V@Rα analysis: an implicit benchmark, one which relates to capital preservation, is the threshold between positive and negative performance, or 0% return. However, in the case of exceptional portfolios of such good expected return and low volatility that provide positive V@Rα measures, what exactly is at risk is no longer clear.
The controversy over V@Rα (and other risk measures) erupted following the fundamental work of P. Artzner, F. Delbaen, J.-M. Eber and D. Heath (see “Thinking Coherently”, Risk 10, November 1997, pp. 68-71 and “Coherent Measures of Risk”, Mathematical Finance, Vol. 9, July 1999, pp. 203-228) and P. Artzner (see “Application of Coherent Risk Measures to Capital Requirements in Insurance”, North American Actuarial Journal, Vol. 3, No. 2, pp. 11-25, 1999). They first distinguish between acceptable and unacceptable risk: a position has unacceptable risk if its future value is unacceptable. A measure of risk of an unacceptable position is then the minimum extra capital that, invested in a reference prudent instrument such as default-free treasury bills, makes the future value of the modified position become acceptable. Risk is thereby described by a real single number or quantity (in effect, the result of mapping of risk functions into the domain of real numbers). In reality, this single number may correspond to the pure insurance premium to be paid out to a secondary market for insurance liabilities or, if such a market does not exist, to the contribution to a reserve built up by the investor to compensate for unacceptable future values. This notion of risk is then consistent with that of the riskless hedge corresponding to the pricing of options for securing investment portfolios, as developed by F. Black and M. Scholes (see “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, Vol. 81, No. 3, May/June 1973, pp. 637-654). U.S. Pat. No. 5,799,287 discusses a computer based method intended to optimize the trade-off between this risk cost and residual profit. P. Artzner, F. Delbaen, J.-M. Eber and D. Heath (see “Thinking Coherently”, Risk 10, November 1997, pp. 68-71 and “Coherent Measures of Risk”, Mathematical Finance, Vol. 9, July 1999, pp. 203-228) continue by setting out four axioms (or self-evident truths that require no proof) that define coherent risk measures. A risk measure satisfying translation invariance, subadditivity, positive homogeneity and monotonicity is called coherent. V@Rα is shown to violate subadditivity: the V@Rα of position 3 obtained from the combination of positions 1 and 2 may be superior to the sum of the V@Rα measures for positions 1 and 2 taken alone. Diversification seemingly leads to an increase in risk, which is incoherent. V@Rα would then dangerously promote concentration, not diversification. A risk measure based simply on a linear combination of the expected return and the variance, standard deviation or semi-variance is also shown to be incoherent.
An important section of the financial community, rightly preoccupied with coherence, has moved quickly to fill the void perceived to be left by V@Rα in promoting worthier risk measures. A measure that has come to the forefront is the Expected Shortfall (or shortfall expectation) at a specified level α, i.e. ESα, with (0≦α≦1) corresponding to a probability of loss as set out by the profit and loss probability density function. ESα is then the mathematical transcript of the concept “average loss in the worst 100α% cases” (see “On the Coherence of Expected Shortfall”, Journal of Banking and Finance, C. Acerbi and D. Tasche, Vol. 26, No. 7, July 2002, pp. 1487-1503). Other similar measures are the Worst Conditional Expectation WCE (see “Thinking Coherently”, P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Risk 10, November 1997, pp. 68-71 and “Coherent Measures of Risk”, Mathematical Finance, Vol. 9, July 1999, pp. 203-228), the Tail Conditional Expectation TCE (or conditional tail expectation CTE or Tail V@R), the Conditional Value-at-Risk CV@R, the α-tail mean and the Mean Excess Loss MEL (see “Extreme Value Theory as a Risk Management Tool”, P. Embrechts, S. I. Resnick and G. Samorodnitsky, North American Actuarial Journal, Vol. 3, No. 2, April 1999, pp. 30-41). All these measures strive essentially to define the same concept but discrepancies may arise in the case of discrete, or mixtures of discrete and continuous probability density functions, if a same quintile a is applicable to more than one threshold, i.e. {P[X≦x]=α} for more than one x. For continuous probability density functions, these measures converge to the same value (see “On the Coherence of Expected Shortfall”, C. Acerbi and D. Tasche, Journal of Banking and Finance, Vol. 26, No. 7, July 2002, pp. 1487-1503).
The acceptance of Expected Shortfall also signifies a convergence between actuaries, statisticians and financial analysts. It is a natural and coherent estimator of risk in a portfolio. To be precise, it refers here to the conditional expected shortfall, i.e. the expected shortfall or most probable shortfall in the event of a shortfall. In that it is preoccupied with measuring the magnitude, severity or intensity of loss given that a loss has occurred, it is a fundamental complement to the measure of frequency or probability of loss. This juxtaposition was also correctly transposed in A. Sen (see “Poverty: an Ordinal Approach to Measurement”, Econometrica, Vol. 44, No. 2, March 1976, pp. 219-231) as pointed out in F. Eggers, A. Pfingsten and S. Rieso (see “Three Dimensions of Shortfall Risk: Transformation and Extension of Sen's Poverty Index”, 9th Symposium on Finance, Banking and Insurance, Universität Karlsruhe (TH), Germany, December 2002).
Various combinations of frequency and severity of investment losses can arise affecting the overall portrayal and classification of risk. Table 15 suggests one such classification based on insurance industry practice. Table 15 is an illustration of the Insurance industry classification of risk based on frequency and severity of loss.
TABLE 15Insurance industry classification of risk based onfrequency and severity of lossFrequency Of LossLowHighSeverityLowNegligibleImportantOf LossTo Very ImportantHighImportantCriticalTo Very Important
U.S. Pat. Nos. 5,884,287, 5,999,918 and 6,012,044 fundamentally rely on point calculations of the probability of loss in providing financial advice.
For those with absolute risk aversion, the only possible solution is risk avoidance attained by investing solely in guaranteed investment products and, ultimately, short term treasury bills.
For those willing to take on elements of risk in their quest for greater returns, risk should be consciously controlled by first establishing rational perspectives with regards to allowable risk levels pertaining to frequency and severity. Continuous monitoring is required to insure these perspectives are met and maintained.